1. Introduction
Variable exponent spaces play an important role in the study of some nonlinear problems in natural science and engineering. In decades, there is a rapid development on the subject of variable exponent function spaces. Basic properties of the spaces have been discussed by Ková
c
˘
ik and Rákosník in [1]. Some theories of variable exponent spaces can also be found in [2, 3]. Harjulehto et al. present an overview of applications to differential equations with nonstandard growth in [4]. Diening et al. [5] summarize most of the existing literature of theory of function spaces with variable exponents and applications to partial differential equations. In [6], Aoyama discusses the properties of Lebesgue spaces with variable exponents on a probability space.
Malliavin derivatives have many applications on mathematical finance (see Malliavin and Thalmaier [7] and di Nunno et al. [8]), and we are interested in the behavior of Malliavin derivatives in spaces with variable exponents. In this paper, in Section 2, motivated by [6, 9, 10], we will first introduce
L
p
(
x
)
(
H
,
μ
)
and
L
p
(
x
)
(
H
,
μ
;
H
)
and give some approximation results of
L
p
(
x
)
(
H
,
μ
)
, which are useful in the definition on Malliavin derivatives in the following parts. In Section 3 we discuss gradients in
L
p
(
x
)
(
H
,
μ
)
and give properties of the linear operator
D
. At the end of this section, we define variable exponent Sobolev spaces on
H
. After the above preparation, we give Malliavin derivatives in
L
p
(
x
)
(
H
,
μ
)
and discuss some properties of the Malliavin derivative operator
M
in
L
p
(
x
)
(
H
,
μ
)
in the last section.
2. Preliminaries
Let
H
be a separable Hilbert space. A Borel probability measure
N
m
,
Q
in
H
with mean
m
and compact covariance operator
Q
is called Gaussian measure if the Fourier transform satisfies
(1)
∫
H
e
i
〈
h
,
x
〉
N
m
,
Q
(
d
x
)
=
e
i
〈
m
,
h
〉
e
(

1
/
2
)
〈
Q
h
,
h
〉
,
∀
x
∈
H
.
N
m
,
Q
is called nondegenerate if
Ker
Q
=
{
0
}
. We are given a nondegenerate Gaussian measure
μ
=
N
0
,
Q
in the Hilbert space
H
. Since the operator
Q
is compact, there exists a complete orthonormal system
{
e
k
}
in
H
and a sequence
{
λ
k
}
of positive numbers such that
(2)
Q
e
k
=
λ
k
e
k
,
k
∈
ℕ
.
We denote by
C
b
(
H
)
the space of all mappings
ϕ
:
H
→
ℝ
, which are both continuous and bounded.
C
b
(
H
)
is a Banach space with the norm
∥
φ
∥
0
=
sup
x
∈
H

φ
(
x
)

. And denote by
C
b
k
(
H
)
,
k
∈
ℕ
, the space of all mappings
ϕ
:
H
→
ℝ
which are continuous and bounded together with their derivatives of order not bigger than
k
.
Given a variable exponent
p
:
H
→
[
1
,
∞
]
, it is assumed to be a Borel measurable function. On the set of all Borel measurable functions, the moduli
ρ
and
ρ
~
are defined, respectively, by
(3)
ρ
(
φ
)
=
∫
H
∖
H
∞

φ
(
x
)

p
(
x
)
μ
(
d
x
)
+
ess
sup
x
∈
H
∞

φ
(
x
)

,
ρ
~
(
F
)
=
∫
H
∖
H
∞

F
(
x
)

H
p
(
x
)
μ
(
d
x
)
+
ess
sup
x
∈
H
∞

F
(
x
)

H
,
where
H
∞
=
{
x
:
p
(
x
)
=
∞
}
,
φ
:
H
→
ℝ
, and
F
:
H
→
H
.
Definition 1.
The space
ℒ
p
(
x
)
(
H
,
μ
)
is the set of Borel measurable functions
φ
:
H
→
ℝ
such that
ρ
(
φ
)
<
∞
, and it is endowed with the Luxemburg norm:
(4)
∥
φ
∥
L
p
(
x
)
(
H
,
μ
)
=
inf
{
λ
>
0
:
ρ
(
φ
λ
)
≤
1
}
.
Define the equivalence relation
r
1
:
φ
~
ψ
if and only if
∥
φ

ψ
∥
L
p
(
x
)
(
H
,
μ
)
=
0
. Denote by
L
p
(
x
)
(
H
,
μ
)
the quotient
ℒ
p
(
x
)
(
H
,
μ
)
with respect to the equivalence relation
r
1
.
Definition 2.
The space
ℒ
p
(
x
)
(
H
,
μ
;
H
)
is the set of Borel measurable functions
F
:
H
→
H
such that
ρ
~
(
F
)
<
∞
, and it is endowed with the Luxemburg norm:
(5)
∥
F
∥
L
p
(
x
)
(
H
,
μ
;
H
)
=
inf
{
λ
>
0
:
ρ
¯
(
F
λ
)
≤
1
}
.
Define the equivalence relation
r
2
:
F
~
G
if and only if
∥
F

G
∥
L
p
(
x
)
(
H
,
μ
;
H
)
=
0
. Denote by
L
p
(
x
)
(
H
,
μ
;
H
)
the quotient
ℒ
p
(
x
)
(
H
,
μ
;
H
)
with respect to the equivalence relation
r
2
.
Proposition 3 (see Lemma <inlineformula>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M64">
<mml:mn>3.2</mml:mn>
<mml:mo>.</mml:mo>
<mml:mn>20</mml:mn></mml:math>
</inlineformula> in [<xref reftype="bibr" rid="B4">5</xref>]).
Let variable exponents
p
(
x
)
,
q
(
x
)
,
r
(
x
)
∈
[
1
,
∞
]
be such that
1
/
r
(
x
)
=
(
1
/
p
(
x
)
)
+
(
1
/
q
(
x
)
)
,
μ
a.e.; then the inequality
(6)
∥
φ
ψ
∥
L
r
(
x
)
(
H
,
μ
)
≤
2
∥
φ
∥
L
p
(
x
)
(
H
,
μ
)
∥
ψ
∥
L
q
(
x
)
(
H
,
μ
)
holds for every
φ
∈
L
p
(
x
)
(
H
,
μ
)
and
φ
∈
L
q
(
x
)
(
H
,
μ
)
, where in the case
r
=
p
=
q
=
∞
we use the convention
r
/
p
=
r
/
q
=
1
.
Proposition 4.
The variable exponent
p
satisfies
1
≤
p

≤
p
(
x
)
≤
p
+
<
∞
,
μ
a.e. Let
F
∈
L
p
(
x
)
(
H
,
μ
;
H
)
and set
F
k
(
x
)
=
〈
F
(
x
)
,
e
k
〉
;
k
∈
ℕ
. Then
F
k
∈
L
p
(
x
)
(
H
,
μ
)
for all
k
∈
ℕ
and
(7)
F
(
x
)
=
∑
k
=
1
∞
F
k
(
x
)
e
k
,
μ

a
.
e
.
,
where the series is convergent in
L
p
(
x
)
(
H
,
μ
;
H
)
.
Proof.
For
F
k
,
k
∈
ℕ
,
(8)
∫
H

F
k
(
x
)

p
(
x
)
μ
(
d
x
)
≤
∫
H

F
(
x
)

H
p
(
x
)
μ
(
d
x
)
<
∞
,
so
F
k
∈
L
p
(
x
)
(
H
,
μ
)
. Since
{
e
k
}
is a complete orthonormal system in
H
and
F
(
x
)
∈
H
for
x
∈
H
, we have
(9)
F
(
x
)
=
∑
k
=
1
∞
F
k
(
x
)
e
k
,
μ
a
.
e
.
Set
P
n
F
(
x
)
=
∑
k
=
1
n
F
k
(
x
)
e
k
, and we have
(10)
lim
n
→
∞
P
n
F
(
x
)
=
F
(
x
)
,
μ
a
.
e
.
,

F
(
x
)

H
p
(
x
)
=
(
∑
k
=
1
∞

F
k
(
x
)

2
)
p
(
x
)
/
2
≥
(
∑
k
=
n
+
1
∞

F
k
(
x
)

2
)
p
(
x
)
/
2
=

P
n
F
(
x
)

F
(
x
)

H
p
(
x
)
,
so by Lebesgue’s Dominated Convergence Theorem, we have
(11)
lim
n
→
∞
∫
H

P
n
F
(
x
)

F
(
x
)

H
p
(
x
)
μ
(
d
x
)
=
0
.
Thus, the series are convergent in
L
p
(
x
)
(
H
,
μ
;
H
)
. The proof is completed.
Denote the linear span of all real and imaginary parts functions
φ
h
(
x
)
=
e
i
〈
h
,
x
〉
,
x
∈
H
, and
h
∈
H
, by
ℰ
(
H
)
.
Proposition 5 (see Lemma 2.2 in [<xref reftype="bibr" rid="B1">9</xref>]).
For all
φ
∈
C
b
(
H
)
there exists a twoindex sequence
{
φ
n
,
k
}
⊂
ℰ
(
H
)
such that
∥
φ
n
,
k
∥
0
≤
∥
φ
∥
0
,
n
,
k
∈
ℕ
, and
(12)
lim
n
→
∞
lim
k
→
∞
φ
n
,
k
(
x
)
=
φ
(
x
)
,
x
∈
H
.
C
c
(
H
)
is the space of all mappings
φ
:
H
→
ℝ
which are continuous and have compact support in
H
. We have the following proposition about
C
c
(
H
)
.
Proposition 6.
If the variable exponent
p
satisfies
1
≤
p

≤
p
(
x
)
≤
p
+
<
∞
,
μ
a.e., then
C
c
(
H
)
is dense in
L
p
(
x
)
(
H
,
μ
)
.
Proof.
Note that
S
≔
S
(
H
,
μ
)
is the set of all the simple functions. It is easy to check that
S
⊂
L
p
(
x
)
(
H
,
μ
)
. Let
f
∈
L
p
(
x
)
(
H
,
μ
)
with
f
≥
0
. Since
f
is a Borel measurable function, there exist
{
f
n
}
⊂
S
with
0
≤
f
n
↗
f
,
μ
a.e., and

f
n
(
x
)

f
(
x
)

p
(
x
)
≤

f
(
x
)

p
(
x
)
, so by Lebesgue's Dominated Convergence Theorem, we have
f
n
→
f
in
L
p
(
x
)
(
H
,
μ
)
. Thus,
f
is in the closure of
S
. If we drop the assumption
f
≥
0
, then we split
f
into positive and negative parts which belong to the closure of
S
. Thus
S
is dense in
L
p
(
x
)
(
H
,
μ
)
. For any
ɛ
>
0
and
g
∈
L
p
(
x
)
(
H
,
μ
)
, there exists
s
∈
S
such that
∥
g

s
∥
L
p
(
x
)
(
H
,
μ
)
<
ɛ
/
2
.
On the other hand, for
p
+
<
∞
,
C
c
(
H
)
is dense in
L
p
+
+
1
(
H
,
μ
)
. So there exists
g
~
∈
C
c
(
H
)
such that
∥
s

g
~
∥
L
p
+
+
1
(
H
,
μ
)
<
ɛ
/
4
. By Proposition 3, we have
(13)
∥
s

g
~
∥
L
p
(
x
)
(
H
,
μ
)
≤
2
∥
s

g
~
∥
L
p
+
+
1
(
H
,
μ
)
<
ɛ
2
.
Thus,
∥
g

g
~
∥
L
p
(
x
)
(
H
,
μ
)
≤
ɛ
and
C
c
(
H
)
is dense in
L
p
(
x
)
(
H
,
μ
)
. The proof is completed.
Proposition 7.
If the variable exponent
p
satisfies
1
≤
p

≤
p
(
x
)
≤
p
+
<
∞
,
μ
a.e., then
ℰ
(
H
)
is dense in
L
p
(
x
)
(
H
,
μ
)
.
Proof.
Since
C
c
(
H
)
⊂
C
b
(
H
)
, by Proposition 6
C
b
(
H
)
is dense in
L
p
(
x
)
(
H
,
μ
)
. So for any
ɛ
>
0
and
f
∈
L
p
(
x
)
(
H
,
μ
)
, there exists
φ
∈
C
b
(
H
)
such that
(14)
∥
f

φ
∥
L
p
(
x
)
(
H
,
μ
)
<
ɛ
2
.
By Proposition 5 and diagonal rule, there exists a sequence
{
φ
n
}
⊂
ℰ
(
H
)
such that

φ
n
(
x
)

≤
∥
φ
∥
0
,
n
∈
ℕ
, and
(15)
lim
n
→
∞
φ
n
(
x
)
=
φ
(
x
)
,
x
∈
H
.
By Lebesgue's Dominated Convergence Theorem, we have
(16)
lim
n
→
∞
∫
H

φ
n
(
x
)

φ
(
x
)

μ
(
d
x
)
=
0
.
And as
(17)
∫
H

φ
n
(
x
)

φ
(
x
)

p
(
x
)
μ
(
d
x
)
≤
(
2
∥
φ
∥
0
)
p
+
∫
H

φ
n
(
x
)

φ
(
x
)

μ
(
d
x
)
,
we have
φ
n
→
φ
in
L
p
(
x
)
(
H
,
μ
)
. Suppose that
N
∈
ℕ
,
∥
φ
n

φ
∥
L
p
(
x
)
(
H
,
μ
)
<
ɛ
/
2
for
n
>
N
, and we have
∥
φ
n

f
∥
L
p
(
x
)
(
H
,
μ
)
<
ɛ
for
n
>
N
. Thus,
ℰ
(
H
)
is dense in
L
p
(
x
)
(
H
,
μ
)
.
Proposition 8.
Suppose that
W
z
,
z
∈
H
, is a white noise function on
H
. If the variable exponent
p
satisfies
1
≤
p

≤
p
(
x
)
≤
p
+
<
∞
,
μ
a.e., then
(18)
∥
W
z
∥
L
p
(
x
)
(
H
,
μ
)
≤
2
(
(
2
m
)
!
2
m
m
!

z

H
m
)
1
/
2
m
,
where
p
+
<
2
m
and
m
∈
ℕ
.
Proof.
Since
W
z
is a Gaussian random variable with mean 0 and covariance

z

H
, by Proposition 3, we have
(19)
∥
W
z
∥
L
p
(
x
)
(
H
,
μ
)
≤
2
∥
1
∥
L
(
2
m
p
(
x
)
)
/
(
2
m

p
(
x
)
)
(
H
,
μ
)
∥
W
z
∥
L
2
m
(
H
,
μ
)
=
2
(
(
2
m
)
!
2
m
m
!

z

H
m
)
1
/
2
m
.
The proof is completed.
3. Variable Exponent Sobolev Spaces on Separable Hilbert Space <inlineformula>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M184">
<mml:mrow>
<mml:mi>H</mml:mi></mml:mrow>
</mml:math></inlineformula>
For any
φ
∈
ℰ
(
H
)
and
k
∈
ℕ
, we denote by
D
k
φ
its derivative in the direction of
e
k
; that is,
(20)
D
k
φ
(
x
)
=
lim
ɛ
→
0
1
ɛ
(
φ
(
x
+
ɛ
e
k
)

φ
(
x
)
)
,
x
∈
H
.
Denote the gradient of
φ
by
D
φ
.
We will consider the following linear mapping:
(21)
D
:
ℰ
(
H
)
⊂
L
p
(
x
)
(
H
,
μ
)
⟶
L
p
(
x
)
(
H
,
μ
;
H
)
,
φ
⟼
D
φ
,
where
1
≤
p

≤
p
(
x
)
≤
p
+
<
∞
,
μ
a.e.
Lemma 9 (see Lemma 2.6 in [<xref reftype="bibr" rid="B1">9</xref>]).
Suppose
φ
,
ψ
∈
ℰ
(
H
)
; then
(22)
∫
H
D
k
φ
ψ
d
μ
=

∫
H
φ
D
k
ψ
d
μ
+
1
λ
k
∫
H
x
k
φ
ψ
d
μ
,
where
x
∈
H
and
x
k
=
〈
x
,
e
k
〉
.
Lemma 10 (see Corollary 2.7 in [<xref reftype="bibr" rid="B1">9</xref>]).
Suppose
φ
,
ψ
∈
ℰ
(
H
)
and
z
∈
Q
1
/
2
(
H
)
; then
(23)
∫
H
〈
D
φ
,
z
〉
ψ
d
μ
=

∫
H
〈
D
ψ
,
z
〉
φ
d
μ
+
∫
H
W
Q

1
/
2
z
φ
ψ
d
μ
.
Given a linear operator
A
, not necessarily closed, if the closure of its graph happens to be the graph of some operator, that operator is called the closure of
A
, and we say that
A
is closable. Denote the closure of
A
by
A
¯
. And
A
is closable if and only if for any sequence
{
x
n
}
⊂
D
(
A
)
such that
(24)
lim
n
→
∞
x
n
=
0
,
lim
n
→
∞
A
x
n
=
y
,
one has
y
=
0
.
Theorem 11.
The mapping
D
is a closable linear operator.
Proof.
Let
{
φ
n
}
⊂
ℰ
(
H
)
such that
(25)
φ
n
⟶
0
in
L
p
(
x
)
(
H
,
μ
)
,
D
φ
n
⟶
F
in
L
p
(
x
)
(
H
,
μ
;
H
)
.
By the definition of closable operators, we only need to prove that
F
=
0
.
For any
ψ
∈
ℰ
(
H
)
and
z
∈
Q
1
/
2
(
H
)
, by Lemma 10, we have
(26)
∫
H
〈
D
φ
n
,
z
〉
ψ
d
μ
=

∫
H
〈
D
ψ
,
z
〉
φ
n
d
μ
+
∫
H
W
Q

1
/
2
z
φ
n
ψ
d
μ
.
By Proposition 3, we have
(27)

∫
H
〈
D
φ
n

F
,
z
〉
ψ
d
μ

≤
∫
H

D
φ
n

F

H

z

H

ψ

H
d
μ
≤
2

z

H
∥
D
φ
n

F
∥
L
p
(
x
)
(
H
,
μ
;
H
)
×
∥
ψ
∥
L
q
(
x
)
(
H
,
μ
)
,

∫
H
〈
D
ψ
,
z
〉
φ
n
d
μ

≤
2

z

H
∥
φ
n
∥
L
p
(
x
)
(
H
,
μ
)
∥
D
ψ
∥
L
q
(
x
)
(
H
,
μ
;
H
)
,
where
(
1
/
p
(
x
)
)
+
(
1
/
q
(
x
)
)
=
1
;
x
∈
H
. Since
ψ
is bounded, by Propositions 3 and 8, we have
(28)

∫
H
W
Q

1
/
2
z
φ
n
ψ
d
μ

≤
2
C
∥
φ
n
∥
L
p
(
x
)
(
H
,
μ
)
∥
W
Q

1
/
2
z
∥
L
q
(
x
)
(
H
,
μ
)
≤
C
p
+

Q

1
/
2
z

H
1
/
2
∥
φ
n
∥
L
p
(
x
)
(
H
,
μ
)
.
Thus,
∫
H
〈
F
,
z
〉
ψ
d
μ
=
0
, as
n
→
∞
. Since
F
k
∈
L
p
(
x
)
(
H
,
μ
)
, for a fixed
k
∈
ℕ
, suppose
{
ψ
k
n
}
⊂
ℰ
(
H
)
such that
ψ
k
n
→

F
k

p
(
x
)

1
sgn
(
F
k
)
in
L
q
(
x
)
(
H
,
μ
)
. We have
(29)
lim
n
→
∞
∫
H
F
k
ψ
k
n
d
μ
=
0
;
that is,
∫
H

F
k

p
(
x
)
d
μ
=
0
. Thus, we have
F
k
(
x
)
=
0
,
μ
a.e., and
F
(
x
)
=
0
,
μ
a.e. The proof is completed.
We denote by
D
¯
the closable operator of
D
and by
W
1
,
p
(
x
)
(
H
,
μ
)
the domain of
D
¯
.
Proposition 12.
For any
k
∈
ℕ
, the linear operator
D
k
is closable and its closure
D
¯
k
satisfies
D
¯
k
φ
=
〈
D
¯
φ
,
e
k
〉
for
φ
∈
W
1
,
p
(
x
)
(
H
,
μ
)
.
Proof.
For any
k
∈
ℕ
, let
{
φ
n
}
⊂
ℰ
(
H
)
such that
(30)
φ
n
⟶
0
in
L
p
(
x
)
(
H
,
μ
)
,
D
k
φ
n
⟶
F
k
in
L
p
(
x
)
(
H
,
μ
)
.
By the definition of closable operators, we only need to prove that
F
k
=
0
. For any
ψ
∈
ℰ
(
H
)
and
z
∈
H
, by Lemma 9, we have
(31)
∫
H
D
k
φ
n
ψ
d
μ
=

∫
H
φ
n
D
k
ψ
d
μ
+
1
λ
k
∫
H
x
k
φ
n
ψ
d
μ
.
Similar to the proof of Theorem 11, we have
∫
H
F
k
ψ
d
μ
=
0
as
n
→
∞
and
F
k
=
0
,
μ
a.e. Thus
D
k
is closable. And
(32)
D
¯
k
φ
=
lim
n
→
∞
D
k
φ
n
=
lim
n
→
∞
〈
D
φ
n
,
e
k
〉
=
〈
D
¯
φ
,
e
k
〉
.
The proof is completed.
4. Malliavin Derivatives in <inlineformula>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M259">
<mml:msup>
<mml:mrow>
<mml:mi>D</mml:mi></mml:mrow>
<mml:mrow>
<mml:mn>1</mml:mn>
<mml:mo mathvariant="bold">,</mml:mo>
<mml:mi>p</mml:mi>
<mml:mo mathvariant="bold">(</mml:mo>
<mml:mi>x</mml:mi>
<mml:mo mathvariant="bold">)</mml:mo></mml:mrow>
</mml:msup>
<mml:mo mathvariant="bold">(</mml:mo>
<mml:mi>H</mml:mi>
<mml:mo mathvariant="bold">,</mml:mo>
<mml:mi>μ</mml:mi>
<mml:mo mathvariant="bold">)</mml:mo></mml:math>
</inlineformula>
For any
φ
∈
ℰ
(
H
)
, define the linear operator:
(33)
M
0
:
ℰ
(
H
)
⊂
L
p
(
x
)
(
H
,
μ
)
⟶
L
p
(
x
)
(
H
,
μ
;
H
)
,
M
0
φ
(
x
)
=
Q
1
/
2
D
φ
(
x
)
,
φ
∈
ℰ
(
H
)
,
x
∈
H
,
where
1
≤
p

≤
p
(
x
)
≤
p
+
<
∞
,
μ
a.e. And
(34)
M
k
φ
(
x
)
=
〈
M
0
φ
(
x
)
,
e
k
〉
=
λ
k
1
/
2
D
k
φ
(
x
)
,
where
φ
∈
ℰ
(
H
)
and
x
∈
H
.
Proposition 13 (see Corollary 2.10 in [<xref reftype="bibr" rid="B1">9</xref>]).
Suppose
φ
,
ψ
∈
ℰ
(
H
)
and
z
∈
H
; then the following identity holds:
(35)
∫
H
〈
M
0
φ
,
z
〉
ψ
d
μ
=

∫
H
〈
M
0
ψ
,
z
〉
φ
d
μ
+
∫
H
W
z
φ
ψ
d
μ
.
In a similar way to Theorem 11, one has Theorem 14.
Theorem 14.
The mapping
M
0
is a closable linear operator.
Proof.
Let
{
φ
n
}
⊂
ℰ
(
H
)
such that
(36)
φ
n
⟶
0
in
L
p
(
x
)
(
H
,
μ
)
,
M
0
φ
n
⟶
F
in
L
p
(
x
)
(
H
,
μ
;
H
)
.
We only need to prove that
F
=
0
.
For any
ψ
∈
ℰ
(
H
)
and
z
∈
H
, by Proposition 13, we have
(37)
∫
H
〈
M
0
φ
n
,
z
〉
ψ
d
μ
=

∫
H
〈
M
0
ψ
,
z
〉
φ
n
d
μ
+
∫
H
W
z
φ
n
ψ
d
μ
.
And we have
∫
H
〈
F
,
z
〉
ψ
d
μ
=
0
, as
n
→
∞
. Similar to Theorem 11, we have
F
(
x
)
=
0
,
μ
a.e. The proof is completed.
We denote by
M
the closable operator of
M
0
and by
D
1
,
p
(
x
)
(
H
,
μ
)
the domain of
M
. For any
φ
∈
D
1
,
p
(
x
)
(
H
,
μ
)
, we call
M
φ
the Malliavin derivative of
φ
.
Proposition 15.
If
p
satisfies
1
≤
p
(
x
)
≤
p
+
<
∞
,
μ
a.e., then
W
1
,
p
(
x
)
(
H
,
μ
)
⊂
D
1
,
p
(
x
)
(
H
,
μ
)
and
M
φ
=
Q
1
/
2
D
¯
φ
for any
φ
∈
W
1
,
p
(
x
)
(
H
,
μ
)
.
Proof.
For any
φ
∈
W
1
,
p
(
x
)
(
H
,
μ
)
, there exists
{
φ
n
}
⊂
ℰ
(
H
)
such that
φ
n
→
φ
in
L
p
(
x
)
(
H
,
μ
)
. As
(38)
∫
H

Q
1
/
2
D
φ
n
(
x
)

Q
1
/
2
D
¯
φ
(
x
)

H
p
(
x
)
μ
(
d
x
)
≤
∫
H
∥
Q
1
/
2
∥
p
(
x
)

D
φ
n
(
x
)

D
¯
φ
(
x
)

H
p
(
x
)
μ
(
d
x
)
≤
max
{
∥
Q
1
/
2
∥
,
∥
Q
1
/
2
∥
p
+
}
∫
H

D
φ
n
(
x
)

D
¯
φ
(
x
)

H
p
(
x
)
μ
(
d
x
)
we have
M
φ
=
Q
1
/
2
D
¯
φ
and
φ
∈
D
1
,
p
(
x
)
(
H
,
μ
)
.
To prove Proposition 17, we need the following lemma.
Lemma 16 (see Lemma 2.3 in [<xref reftype="bibr" rid="B1">9</xref>]).
For all
φ
∈
C
b
1
(
H
)
there exists a twoindex sequence
{
φ
n
,
k
}
⊂
ℰ
(
H
)
such that
∥
φ
n
,
k
∥
0
+
∥
D
φ
n
,
k
∥
0
≤
∥
φ
∥
0
+
∥
D
φ
∥
0
,
n
,
k
∈
ℕ
, and
(39)
lim
n
→
∞
lim
k
→
∞
φ
n
,
k
(
x
)
=
φ
(
x
)
,
x
∈
H
,
lim
n
→
∞
lim
k
→
∞
D
φ
n
,
k
(
x
)
=
D
φ
(
x
)
,
x
∈
H
.
Proposition 17.
If
p
satisfies
1
≤
p
(
x
)
≤
p
+
<
∞
,
μ
a.e., then one has
g
(
φ
)
∈
D
1
,
p
(
x
)
(
H
,
μ
)
and
M
g
(
φ
)
=
g
′
(
φ
)
M
φ
for any
φ
∈
D
1
,
p
(
x
)
(
H
,
μ
)
and
g
∈
C
b
1
(
H
)
.
Proof.
First, we prove that
g
(
ψ
)
∈
D
1
,
p
(
x
)
(
H
,
μ
)
for any
ψ
∈
ℰ
(
H
)
. By Lemma 16, for
g
(
ψ
)
, there exists a twoindex sequence
{
ϕ
n
,
k
}
⊂
ℰ
(
H
)
such that
∥
ϕ
n
,
k
∥
0
+
∥
D
ϕ
n
,
k
∥
0
≤
∥
g
(
ψ
)
∥
0
+
∥
D
g
(
ψ
)
∥
0
,
n
,
k
∈
ℕ
, and
(40)
lim
n
→
∞
lim
k
→
∞
ϕ
n
,
k
(
x
)
=
g
(
ψ
)
(
x
)
,
x
∈
H
,
lim
n
→
∞
lim
k
→
∞
D
ϕ
n
,
k
(
x
)
=
D
g
(
ψ
)
(
x
)
,
x
∈
H
.
By diagonal rule and Lebesgue’s Dominated Convergence Theorem,
ϕ
n
→
g
(
ψ
)
in
L
p
(
x
)
(
H
,
μ
)
. Since
M
is closed, we have
M
g
(
ψ
)
=
lim
n
→
∞
M
0
ϕ
n
and
g
(
ψ
)
∈
D
1
,
p
(
x
)
(
H
,
μ
)
.
Now we will prove the proposition. For
φ
∈
D
1
,
p
(
x
)
(
H
,
μ
)
, there exists
{
φ
n
}
⊂
ℰ
(
H
)
such that
(41)
lim
n
→
∞
φ
n
=
φ
in
L
p
(
x
)
(
H
,
μ
)
.
By the first part of this proof,
g
(
φ
n
)
∈
D
1
,
p
(
x
)
(
H
,
μ
)
. And
(42)
M
g
(
φ
n
)
=
Q
1
/
2
D
g
(
φ
n
)
=
g
′
(
φ
n
)
Q
1
/
2
D
φ
n
=
g
′
(
φ
n
)
M
φ
n
.
Since
M
φ
n
∈
L
p
(
x
)
(
H
,
μ
;
H
)
and
g
′
(
φ
n
)
∈
C
b
(
H
)
, we have
(43)
lim
n
→
∞
M
g
(
φ
n
)
=
lim
n
→
∞
g
′
(
φ
n
)
M
φ
n
=
g
′
(
φ
)
M
φ
.
As
M
is closed,
M
g
(
φ
)
=
g
′
(
φ
)
M
φ
. The proof is completed.
Proposition 18.
Assume that
p
satisfies
1
≤
p
(
x
)
≤
p
+
<
∞
,
μ
a.e. For
φ
,
ψ
∈
D
1
,
p
(
x
)
(
H
,
μ
)
; suppose
ψ
and
M
ψ
are bounded; then
φ
ψ
∈
D
1
,
p
(
x
)
(
H
,
μ
)
and
M
(
φ
ψ
)
=
ψ
M
φ
+
φ
M
ψ
.
Proof.
First, for
φ
∈
ℰ
(
H
)
, as
ψ
∈
D
1
,
p
(
x
)
(
H
,
μ
)
, there exists
{
ψ
n
}
⊂
ℰ
(
H
)
such that
ψ
n
→
ψ
in
L
p
(
x
)
(
H
,
μ
)
. And since
(44)
M
0
(
φ
ψ
n
)
=
ψ
n
M
0
φ
+
φ
M
0
ψ
n
,
we have
(45)
lim
n
→
∞
M
0
(
φ
ψ
n
)
=
lim
n
→
∞
(
ψ
n
M
0
φ
+
φ
M
0
ψ
n
)
=
ψ
M
0
φ
+
φ
M
ψ
.
As
φ
ψ
n
∈
ℰ
(
H
)
,
φ
ψ
n
→
φ
ψ
in
L
p
(
x
)
(
H
,
μ
)
, and
M
(
φ
ψ
)
=
lim
n
→
∞
M
0
(
φ
ψ
n
)
in
L
p
(
x
)
(
H
,
μ
;
H
)
, we have
φ
ψ
∈
D
1
,
p
(
x
)
(
H
,
μ
)
.
Secondly, for
φ
∈
D
1
,
p
(
x
)
(
H
,
μ
)
, there exists
{
φ
n
}
⊂
ℰ
(
H
)
such that
φ
n
→
φ
in
L
p
(
x
)
(
H
,
μ
)
. By the first part of the proof,
(46)
M
(
φ
n
ψ
)
=
ψ
M
0
φ
n
+
φ
n
M
ψ
.
Since
ψ
and
M
ψ
are bounded, we have
(47)
M
(
φ
ψ
)
=
lim
n
→
∞
M
0
(
φ
n
ψ
)
=
ψ
M
φ
+
φ
M
ψ
,
and
φ
ψ
∈
D
1
,
p
(
x
)
(
H
,
μ
)
. The proof is completed.